Problem 15
Question
Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$B-C$$
Step-by-Step Solution
Verified Answer
Matrix subtraction \(B - C\) is defined and can be computed if and only if \(p = r\) and \(q = s\). When the operation is defined, the size of the resulting matrix is \(p \times q\).
1Step 1: Check for dimensions
First, let's look at the dimensions of both matrices B and C:
- Matrix B is \(p \times q\)
- Matrix C is \(r \times s\)
2Step 2: Look for the conditions for subtraction of matrices
The matrix subtraction operation is defined if and only if the dimensions of both matrices are the same.
For the subtraction \(B - C\) to be valid, the dimensions of B and C must match. Thus, we need to have:
\(p = r\) and \(q = s\)
3Step 3: Find the size of the resulting matrix#
\
If the matrix subtraction is valid, then the size of the resulting matrix is the same as the dimensions of the original matrices:
Matrix B-C will have a size of \(p \times q\)
To summarize, matrix subtraction \(B - C\) is defined and can be computed if and only if \(p = r\) and \(q = s\). When the operation is defined, the size of the resulting matrix is \(p \times q\).
Key Concepts
Matrix DimensionsMatrix OperationsConditions for Matrix Subtraction
Matrix Dimensions
Understanding matrix dimensions is crucial for performing matrix operations. The dimensions of a matrix are specified by the number of rows and columns it contains, typically noted as an 'm x n' format. For instance, if a matrix has 3 rows and 5 columns, its dimensions are expressed as 3 x 5.
When you're given two matrices, like matrix B with dimensions (p x q) and matrix C with dimensions (r x s), these dimensions inform you how many entries the matrices have and their structure. It's like knowing the size of a digital image: the number of pixels in each dimension tells you how large the image is and whether two images can align perfectly.
When you're given two matrices, like matrix B with dimensions (p x q) and matrix C with dimensions (r x s), these dimensions inform you how many entries the matrices have and their structure. It's like knowing the size of a digital image: the number of pixels in each dimension tells you how large the image is and whether two images can align perfectly.
Matrix Operations
Matrix operations include addition, subtraction, multiplication, and division (the latter often through the multiplication of a matrix inverse). To perform these operations, certain conditions about the dimensions must be satisfied.
For example, in matrix addition and subtraction, every element of one matrix is added to or subtracted from the corresponding element of another matrix. Thus, these operations are only possible when both matrices have the same 'shape' - that is, the same number of rows and columns. This similarity in structure is what makes the operation computationally feasible and meaningful.
For example, in matrix addition and subtraction, every element of one matrix is added to or subtracted from the corresponding element of another matrix. Thus, these operations are only possible when both matrices have the same 'shape' - that is, the same number of rows and columns. This similarity in structure is what makes the operation computationally feasible and meaningful.
Conditions for Matrix Subtraction
Moving on to the specific case of matrix subtraction, for which the exercise provided serves as an example, the operation can only be performed when the matrices involved have identical dimensions. In other words, each matrix must have the same number of rows (p must equal r) and columns (q must equal s).
If the dimensions do not match, the subtraction operation is not defined because there is no one-to-one correspondence between the elements of the two matrices. When the condition is met, and you can subtract matrix C from matrix B, the resulting matrix will have the same dimensions as the original matrices, which in our exercise would be a (p x q) matrix. This rule maintains the integrity of the matrix structure and ensures that the operation produces a matrix result that is meaningful in the context of the given matrices.
If the dimensions do not match, the subtraction operation is not defined because there is no one-to-one correspondence between the elements of the two matrices. When the condition is met, and you can subtract matrix C from matrix B, the resulting matrix will have the same dimensions as the original matrices, which in our exercise would be a (p x q) matrix. This rule maintains the integrity of the matrix structure and ensures that the operation produces a matrix result that is meaningful in the context of the given matrices.
Other exercises in this chapter
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